Questions tagged [billiards]

For questions about billiards; dynamical systems involving point particles (billiard balls) that travel in straight lines on the interior of some bounded domain (the billiard table) and experience perfectly elastic reflections on collision with boundary.

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Arithmetic billiards, prime numbers and the Goldbach conjecture

On the online encyclopedia Wikipedia are edited the articles Arithmetic billiards and Dedekind psi function, the online encyclopedia Wolfram MathWorld has edited the article Goldbach Conjecture. We ...
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7 votes
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Why a measure over the points of a billiard which reflect infinitely often in finite time vanish, i.e. why $\int_{N\cap M}f(x^{-})d\mu_1(x^{-1})=0$

The crux of my question is that why $\mu(N^{(2)}) = 0$ when $N^{(2)}$ consists all points of a billiard that reflect infinitely often in finite time under a given flow (i.e. transformation) function. ...
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For arbitrary billiard tables with elastic boundary reflections, is the "Lebesgue measure" an invariant of the flow maps?

This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) ...
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Particular values for the sum of divisors function from billiards

In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article. I've wondered if we can compute some simple ...
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Motivation for the definition of a compact Riemann manifold with piecewise smooth boundary

I am asking for either motivation on the requirement regarding $f_i^{-1}(0)$ in the following definition, or better yet a reference to a book dealing with this subject. The following is the definition ...
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Plotting points which get further away when given the distance and amount of plot points

I am trying to mark up my pool table to run a cutting drill but I don't understand how to do the math which would give me the solution for my plot points. I have a total of 15 marks I need to make on ...
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3 votes
1 answer
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Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit

In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is: How can I reproduce Schwartz's result ...
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2 votes
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Proving a billiards bank shot procedure gives equal angles of incidence and reflection

One method of making a bank shot in billiards involves imagining two lines, the so called "cross-pocket" and "cross-ball" lines: One then projects their point of intersection to ...
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1 vote
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Law of reflection when tangent is undefined

The law of reflection also holds for non-plane mirrors, provided that the normal at any point on the mirror is understood to be the outward pointing normal to the local tangent plane of the mirror at ...
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4 votes
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Application of billiards

Studying billiards is a difficult problem in general, even in pretty simple cases it has plenty of interesting properties. I would like to understand what can be applications (mathematically or in ...
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Rectangular billiards: ergodic or not?

I'm approaching dynamical billiards theory, and I've found that rectangular/square billiards are among the examples of integrable and non-ergodic ones. But in this contribution it is (resolutely) said ...
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Sinai billiard isomorphic to hard spheres problem

I have heard that Sinai's famous results on the ergodicity of $n$ hard spheres defined on a $d$ dimensional hypercube with periodic boundaries is isomorphic to a billiard (from which it was proved), ...
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Phase space and collision map of a billiard

I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions. 2.5. Phase space for the flow. The state of a ...
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Number of rays intersecting at a point inside a polygon

I'm working on a project to do with bouncing rays inside polygons and now I've reached a crucial stage of this project in which I need help with and is related to the problem stated below. Your help ...
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Billiards in an ellipse -- a query from a proof

I am reading about billiards in an ellipse and am a little stuck on a couple of points in the following proof (see image). My main doubt is the highlighted line at the bottom: "Notice that line $...
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Why is the existence of periodic orbits (in triangular billiards) so hard to prove?

I don't get why proving that every triangular billiard has a periodic orbit should be that hard. I mostly understand the partial results on the matter, mainly Every triangle with rational angles ...
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8 votes
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Bouncing Bullet Problem

This is a problem that was presented to me through the google foobar challenge, but my time has since expired and they had decided that I did not complete the problem. I suspect a possible bug on ...
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5 votes
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Do these balls collide?

Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when ...
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4 votes
1 answer
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How to show that the Billiard flow is invariant with respect to the area form $\sin(\alpha)d\alpha\wedge dt$

Consider a plane billiard table $D \subset \mathbb{R}^2$ (i.e. a bounded open connected set) with smooth boundary $\gamma$ being a closed curve. Next, let $M$ denote the space of tangent unit vectors $...
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Texts on Mathematical Billiards

I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains ...
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Verifying that a billiard system satisfies the twist condition.

I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre. ...
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Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
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5 votes
1 answer
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Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
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1 vote
1 answer
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Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
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1 answer
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Joining boundary points through affine/billiards trajectories

Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path ...
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3 votes
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Billiard ball mental patients.

The Question: Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital. Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$ metres ...
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Unfolding a billiard trajectory on an unbounded billiard table

A common "trick" used in the study of polygonal billiards is to unfold the trajectory. I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, ...
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Reflecting a vector within a box (variation on the billiards problem)

Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers. A starting point is chosen at random within the box at (x,y), such ...
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4 votes
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The simplest billiards problem

The Problem: Let's say we have a rectangle of size $m \times n$ centered at the origin (or, if it makes the math easier, you can place it wherever on the plane). We take a billiard ball, ...
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Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
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Trajectories in circular Billards

Given two points $p_1,p_2$ on a circular billard table. I want to know all billard trajectories from $p_1$ to $p_2$ hitting the boundary precisely once. Model of the circular billard: Denote the ...
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3 votes
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Natural projection of the billiard phase space

Setup The phase space of the billiard flow $\left\lbrace \Phi^t \right\rbrace$ ($t \in \mathbb{R}$) is given by $\Omega = \mathcal{D} \times S^1$ where $\mathcal{D}$ is a planar billiard table in $\...
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About a theorem by Mather on smooth convex billiards

I'm a masters student and in about a week I begin working in a short presentation of "a theorem" by John N. Mather. More precisely, I have to give some elements of the proof of a theorem that states ...
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Infinitely many triangular numbers which are of form $n^2-1$

After playing a round of pool billiard recently, I noticed that the fifteen colored balls can be arranged in a triangle (obviously) and all sixteen balls (including the white ball) can be arranged in ...
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6 votes
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Launching billiard balls at 45 degree angles and bouncing of off edges.

Say I have a billiard ball and I launch it from the bottom-left corner of a table with length $x$ and width $y$. Given $x$ and $y$ will the ball reach a corner again, which corner and in how many ...
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2 votes
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Number of possible rays in rectangular mirror room between two points

Question There is a rectangular room with mirrors of $a$ width and $b$ height (Aligned with co-ordinate axes, so lower left corner is at $(0, 0)$ and upper right corner is at $(a, b)$). There are two ...
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2 votes
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Quantitative estimates on space filling curves

To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
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2 votes
1 answer
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Tangent to hyperbola meeting on ellipse

Can anyone prove that the two tangent lines to hyperbola that meet on a confocal ellipse make the same angle with the tangent of the ellipse at the point of intersection? Thanks
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Area of a billiard and its preservation

From S. Tabachnikov's Geometry and billiards Consider a plane billiard table $D$ whose boundary is a smooth closed curve $γ$. Let $M$ be the space of unit tangent vectors $(x, v)$ whose foot ...
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About trajectories on a circular billiard

Let consider a circular billiard. You start by putting the ball on the edge off the table. The trajectory then is quite simple (see the picture), with equal angles at each rebound. Which will ...
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Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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Billiards in a circular table

This is a variation of Alhazen's Billiard Problem. Suppose we have a semicircular billiards table of radius r centered at the origin O, and a billiard ball placed somewhere on the 'x-axis' of the ...
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3 votes
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Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
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Is there an elementary proof for the envelope of an infinitely-bouncing billiard ball?

I was reading this page here http://cage.ugent.be/~hs/billiards/billiards.html And was wondering if there existed an elementary proof of proving that the envelopes form certain conic sections (ie: ...
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2 votes
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For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?

I'm looking for a kind of generalisation of an ellipse; a shape with a more complicated optical property. I'm not sure how to rigorously define this shape, or prove that it exists, or find an equation ...
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0 votes
1 answer
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Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
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1 vote
2 answers
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Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
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Constants of motion for non-chaotric billiards

As I understand it, for a mechanical system, each symmetry leads to a constant of motion. For integrable systems, the number of constants of motion equals the number of degrees of freedom. So take 2D ...
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Turning a rhombus billiard into an equivalent barrier billiard

I am reading a research paper and the authors map the rhombus billiard (angles $60$-$120$ degrees) to an equivalent barrier billiard. They start with a rhombus standing upright and reflect in a ...
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3 votes
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Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
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